DOCSLIB.ORG

21 Non-Linear Wave Equations and Solitons

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
21 Non-Linear Wave Equations and Solitons Utah State University DigitalCommons@USU Foundations of Wave Phenomena Open Textbooks 8-2014 21 Non-Linear Wave Equations and Solitons Charles G. Torre Department of Physics, Utah State University, [email protected] Follow this and additional works at: https://digitalcommons.usu.edu/foundation_wave Part of the Physics Commons To read user comments about this document and to leave your own comment, go to https://digitalcommons.usu.edu/foundation_wave/2 Recommended Citation Torre, Charles G., "21 Non-Linear Wave Equations and Solitons" (2014). Foundations of Wave Phenomena. 2. https://digitalcommons.usu.edu/foundation_wave/2 This Book is brought to you for free and open access by the Open Textbooks at DigitalCommons@USU. It has been accepted for inclusion in Foundations of Wave Phenomena by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. Foundations of Wave Phenomena, Version 8.2 21. Non-linear Wave Equations and Solitons. In 1834 the Scottish engineer John Scott Russell observed at the Union Canal at Hermiston a well-localized* and unusually stable disturbance in the water that propagated for miles virtually unchanged. The disturbance was stimulated by the sudden stopping of a boat on the canal. He called it a “wave of translation”; we call it a solitary wave. As it happens, a number of relatively complicated – indeed, non-linear – wave equations can exhibit such a phenomenon. Moreover, these solitary wave disturbances will often be stable in the sense that if two or more solitary waves collide then after the collision they will separate and take their original shape. Solitary waves which have this stability property are called solitons. The terminology stems from a combination of the word solitary and the suffix “on” which is used to signify a particle (think of the proton, electron, neutron, etc. ). We shall discuss a little later the sense in which a soliton is like a particle. Solitary waves and solitons have become very important in a variety of physical settings, for example: hydrodynamics, non-linear optics, plasmas, meteorology, and elementary particle physics, to name a few. Our goal in this chapter is to give a very brief — and completely superficial — introduction to solitonic solutions of non-linear wave equations. To begin, let me point out that the humble wave equation in one dimension already provides an illustration of some of the phenomena we want to explore, principally by virtue of its linearity. We have already seen that the solutions to the wave equation † @2 @2 v2 q(x, t) = 0 (21.1) @t2 − @x2 ✓ ◆ take the general form q(x, t)=f(x + vt)+g(x vt), (21.2) − where the functions f and g are determined by initial conditions. Let us suppose that we choose our initial conditions so that the solution has f = 0, so that the wave is simply the displacement profile q = g(x vt), that is, a traveling disturbance with the shape dictated − by the curve y = g(x) translating rigidly to the “right” (toward positive x) at speed v. Let us also suppose that g is a function that is localized in some region, so that it has a finite width. We have a “pulse”, which travels to the right, unchanged in shape. Thus the pulse is a solitary wave. To visualize this, imagine that you and your friend are holding a rope taut between you and you shake the end of the rope one time. The result is a pulse which travels toward your friend (with a speed depending upon the density and tension of the rope). This “pulse” has its shape described by the function g. Now suppose we also allow f to be a localized non-trivial function, you then get a pulse traveling to the * About 10 meters long and half a meter in height. The truly remarkable thing about solitonic behavior is that there are highly non-linear † equations which also can exhibit it. 171 c C. G. Torre Foundations of Wave Phenomena, Version 8.2 left superposed with the pulse traveling to the right. Suppose the two pulses are initially separated, the one described by f sitting o↵ at large, positive x, and the one described by g sitting o↵ at large negative x, say. The pulses will approach each other at a relative speed of 2v and at some point they will overlap, giving a wave profile which is, evidently, the algebraic sum of the individual pulses. Eventually, the pulses will become separated with the pulse described by g moving o↵ toward large positive values of x and the pulse described by f moving to large negative values of x. The pulses “collide”, but after the collision they retain their shape – their “identity”, if you will. To visualize this, return to the rope experiment. You shake the rope once, and your friend also shakes the rope once. Each of you produce a pulse which travels toward the other person, overlap for a time, then separate again unscathed. Try it! (You may need a long rope to see this work.) This is a simple example of solitonic behavior. In this example the solitonic behavior results from the linearity of the wave equation (superposition!) and its dispersion relation. Indeed, it is difficult to imagine such behavior emerging from anything but a linear equation. Our model of coupled oscillators that led to the wave equation is about as simple as it can be. As such, it does not take into account many details of the behavior of the medium that the waves propagate in. More realistic or alternate wave equations do not necessarily exhibit this solitary wave behavior because they lack the superposition property and or they lack the requisite dispersion relation. For example, let us consider the Schr¨odinger equation for a free, non-relativistic, quan- tum mechanical particle with mass m moving in one dimension: @ ¯h2 @2 i¯h = . (21.3) @t −2m @x2 This equation, being linear, respects the superposition property, but you will recall it does not have the simple dispersion relation possessed by the wave equation. As we saw, the general solution of (21.3) is a superposition of traveling waves 1 1 i(kx !(k)t) (x, t)= C(k)e − dk, (21.4) p2⇡ Z1 where C(k) is determined by a choice of initial wave function, (x, 0), and where ¯hk2 !(k)= . (21.5) 2m The traveling waves appearing in the superposition have di↵erent speeds. To see this, just note that the wave with wavenumber k has a speed given by !(k) ¯hk v(k)= = . (21.6) k 2m The consequence of this dispersion relation is that at any time t = 0 a well-localized initial 6 wave function does not retain its shape. Indeed, the wave pulse will spread as time passes 172 c C. G. Torre Foundations of Wave Phenomena, Version 8.2 because its di↵erent frequency (or wavelength) components do not travel at the same speed. (Physically, this is a manifestation of the uncertainty principle: localizing the particle at t = 0 within the support of the initial pulse leads to an uncertainty in momentum which, at later times, reduces the localization of the particle.) One says that the Schr¨odinger waves exhibit dispersion because the wave profiles “disperse” as time runs. So we see that solitary wave behavior is not a universal feature of wave phenomena. Let us look at another linear wave equation that exhibits dispersion. The following equation is known as the Klein-Gordon equation (in one spatial dimension): 1 @2 @2 q(x, t)+m2c2q(x, t)=0. (21.7) c2 @t2 − @x2 ✓ ◆ This equation can be used to describe a relativistic quantum particle with rest mass de- termined by m (moving in one dimension). In this context, c is the speed of light – it is not the speed of the waves. You can think of it as a relativistic generalization of the Schr¨odinger equation. Just like the Schr¨odinger equation, the general solution of the Klein-Gordon equation can be constructed by superimposing sinusoidal wave solutions over amplitudes, phases and wavelengths. To see that dispersion arises, we simply compute the dispersion relation that arises for a sinusoidal wave. Consider a (complex) solution of the form i(kx !t) q(x, t)=Ae − . (21.8) It is not hard to see that this wave solves the Klein-Gordon equation if and only if !2 = c2k2 + m2c4 ! = c k2 + m2c2. (21.9) () ± p As you can see, the wave speed !/k again depends upon k, leading to dispersion. Evidently, dispersion in linear wave equations does not allow for solitary wave phenom- ena. Remarkably, one can compensate for dispersion by carefully altering the superposition property using non-linearities in the wave equation. A detailed study of non-linear wave equations is way beyond the scope of this text. My plan is to just have a quick at one, relatively simple non-linear partial di↵erential equation to get a glimpse of how solitons can arise. A relatively simple non-linear equation is given by a modification of the Klein-Gordon equation (in one spatial dimension) for a scalar field φ(x, t): @2φ @2φ m3 pλ + sin( φ)=0. (21.10) @t2 − @x2 pλ m For simplicity in what follows we have chosen units in which c = 1. The presence of the non-linearity is controlled by m.Ifm = 0 we have the usual wave equation. The new 173 c C. G. Torre Foundations of Wave Phenomena, Version 8.2 parameter λ is going to be a “self-coupling constant”. The equation (21.10) is sometimes called the “sine-Gordon equation”, showing that mathematicians have a sense of humor to some extent.
Load more

Recommended publications
  • Waves on Deep Water, I
    Lecture 14: Waves on deep water, I Lecturer: Harvey Segur. Write-up: Adrienne Traxler June 23, 2009 1 Introduction In this lecture we address the question of whether there are stable wave patterns that propagate with permanent (or nearly permanent) form on deep water. The primary tool for this investigation is the nonlinear Schr¨odinger equation (NLS). Below we sketch the derivation of the NLS for deep water waves, and review earlier work on the existence and stability of 1D surface patterns for these waves. The next lecture continues to more recent work on 2D surface patterns and the effect of adding small damping. 2 Derivation of NLS for deep water waves The nonlinear Schr¨odinger equation (NLS) describes the slow evolution of a train or packet of waves with the following assumptions: • the system is conservative (no dissipation) • then the waves are dispersive (wave speed depends on wavenumber) Now examine the subset of these waves with • only small or moderate amplitudes • traveling in nearly the same direction • with nearly the same frequency The derivation sketch follows the by now normal procedure of beginning with the water wave equations, identifying the limit of interest, rescaling the equations to better show that limit, then solving order-by-order. We begin by considering the case of only gravity waves (neglecting surface tension), in the deep water limit (kh → ∞). Here h is the distance between the equilibrium surface height and the (flat) bottom; a is the wave amplitude; η is the displacement of the water surface from the equilibrium level; and φ is the velocity potential, u = ∇φ.
    [Show full text]
  • Shallow Water Waves and Solitary Waves Article Outline Glossary
    Shallow Water Waves and Solitary Waves Willy Hereman Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado, USA Article Outline Glossary I. Definition of the Subject II. Introduction{Historical Perspective III. Completely Integrable Shallow Water Wave Equations IV. Shallow Water Wave Equations of Geophysical Fluid Dynamics V. Computation of Solitary Wave Solutions VI. Water Wave Experiments and Observations VII. Future Directions VIII. Bibliography Glossary Deep water A surface wave is said to be in deep water if its wavelength is much shorter than the local water depth. Internal wave A internal wave travels within the interior of a fluid. The maximum velocity and maximum amplitude occur within the fluid or at an internal boundary (interface). Internal waves depend on the density-stratification of the fluid. Shallow water A surface wave is said to be in shallow water if its wavelength is much larger than the local water depth. Shallow water waves Shallow water waves correspond to the flow at the free surface of a body of shallow water under the force of gravity, or to the flow below a horizontal pressure surface in a fluid. Shallow water wave equations Shallow water wave equations are a set of partial differential equations that describe shallow water waves. 1 Solitary wave A solitary wave is a localized gravity wave that maintains its coherence and, hence, its visi- bility through properties of nonlinear hydrodynamics. Solitary waves have finite amplitude and propagate with constant speed and constant shape. Soliton Solitons are solitary waves that have an elastic scattering property: they retain their shape and speed after colliding with each other.
    [Show full text]
  • Soliton and Some of Its Geophysical/Oceanographic Applications
    Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Soliton and Some of its Geophysical/Oceanographic Applications M. Lakshmanan Centre for Nonlinear Dynamics Bharathidasan University Tiruchirappalli – 620024 India Discussion Meeting, ICTS-TIFR 21 July 2016 M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Theme Soliton is a counter-intuitive wave entity arising because of a delicate balance between dispersion and nonlinearity. Its major attraction is its localized nature, finite energy and preservation of shape under collision = a remark- ably stable structure. Consequently, it has⇒ ramifications in as wider areas as hydrodynamics, tsunami dynamics, condensed matter, magnetism, particle physics, nonlin- ear optics, cosmology and so on. A brief overview will be presented with emphasis on oceanographic applications and potential problems will be indicated. M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory Applications Higher dimensions Problems Plan 1 Introduction 2 Historical: (i) Scott-Russel (ii) Zabusky/Kruskal 3 Korteweg-de Vries equation and solitary wave/soliton 4 Other ubiquitous equations 5 Mathematical theory of solitons 6 Geophysical/Oceanographic applications 7 Solitons in higher dimensions 8 Problems and potentialities M. Lakshmanan Soliton and Some of its Geophysical Introduction Historical KdV equation Other equations Theory
    [Show full text]
  • Dissipative Cnoidal Waves (Turing Rolls) and the Soliton Limit in Microring Resonators
    Research Article Vol. X, No. X / Febrary 1046 B.C. / OSA Journal 1 Dissipative cnoidal waves (Turing rolls) and the soliton limit in microring resonators ZHEN QI1,*,S HAOKANG WANG1,J OSÉ JARAMILLO-VILLEGAS2,M INGHAO QI3, ANDREW M. WEINER3,G IUSEPPE D’AGUANNO1,T HOMAS F. CARRUTHERS1, AND CURTIS R. MENYUK1 1University of Maryland at Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA 2Technological University of Pereira, Cra. 27 10-02, Pereira, Risaralda 660003, Colombia 3Purdue University, 610 Purdue Mall, West Lafayette, IN 47907, USA *Corresponding author: [email protected] Compiled May 27, 2019 Single solitons are a special limit of more general waveforms commonly referred to as cnoidal waves or Turing rolls. We theoretically and computationally investigate the stability and acces- sibility of cnoidal waves in microresonators. We show that they are robust and, in contrast to single solitons, can be easily and deterministically accessed in most cases. Their bandwidth can be comparable to single solitons, in which limit they are effectively a periodic train of solitons and correspond to a frequency comb with increased power. We comprehensively explore the three-dimensional parameter space that consists of detuning, pump amplitude, and mode cir- cumference in order to determine where stable solutions exist. To carry out this task, we use a unique set of computational tools based on dynamical system theory that allow us to rapidly and accurately determine the stable region for each cnoidal wave periodicity and to find the instabil- ity mechanisms and their time scales. Finally, we focus on the soliton limit, and we show that the stable region for single solitons almost completely overlaps the stable region for both con- tinuous waves and several higher-periodicity cnoidal waves that are effectively multiple soliton trains.
    [Show full text]
  • Three-Dimensional Dynamics of Baroclinic Tides Over a Seamount
    University of Plymouth PEARL https://pearl.plymouth.ac.uk Faculty of Science and Engineering School of Biological and Marine Sciences 2018-02 Three-dimensional dynamics of baroclinic tides over a seamount Vlasenko, V http://hdl.handle.net/10026.1/10845 10.1002/2017JC013287 Journal of Geophysical Research: Oceans American Geophysical Union All content in PEARL is protected by copyright law. Author manuscripts are made available in accordance with publisher policies. Please cite only the published version using the details provided on the item record or document. In the absence of an open licence (e.g. Creative Commons), permissions for further reuse of content should be sought from the publisher or author. JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1002/, Three-dimensional dynamics of baroclinic tides over a seamount 1 1 1 Vasiliy Vlasenko , Nataliya Stashchuk , and W. Alex M. Nimmo-Smith Vasiliy Vlasenko, School of Biological and Marine Sciences, Plymouth University, Plymouth, PL4 8AA, UK ([email protected]) 1School of Biological and Marine Sciences, Plymouth University, Plymouth, PL4 8AA, UK. D R A F T January 17, 2018, 10:45am D R A F T X-2 VLASENKO ET AL.: BAROCLINIC TIDES OVER A SEAMOUNT 1 Abstract. 2 The Massachusetts Institute of Technology general circulation model is used 3 for the analysis of baroclinic tides over Anton Dohrn Seamount (ADS), in 4 the North Atlantic. The model output is validated against in-situ data col- 5 lected during the 136-th cruise of the RRS “James Cook” in May-June 2016. 6 The observational data set includes velocity time series recorded at two moor- 7 ings as well as temperature, salinity and velocity profiles collected at 22 hy- 8 drological stations.
    [Show full text]
  • Plasmon-Soliton
    Plasmon-Soliton Eyal Feigenbaum and Meir Orenstein Department of Electrical Engineering, Technion, Haifa 32000, Israel [email protected] Abstract : Formation of a novel hybrid-vector spatial plasmon-soliton in a Kerr slab embedded in-between metal plates is predicted and analyzed with a modified NLSE, encompassing hybrid vector field characteristics. Assisted by the transverse plasmonic effect, the self trapping dimension of the plasmon-soliton was substantially compressed (compared to the dielectrically cladded slab case) when reducing the slab width. The practical limitation of the Plasmon-soliton size reduction is determined by available nonlinear materials and metal loss. For the extreme reported values of nonlinear index change, we predict soliton with a cross section of 300nm×30nm (average dimension of 100nm). 1 The downscaling of conventional photonics is halted when approaching a transverse dimension of ~ λ/2 ( λ is the wavelength). This limitation can be alleviated by the incorporation of metals, giving rise to surface plasmon polaritons (SPP) [1,2]. Here we analyze a novel configuration where light is transversely tightly guided between two metal layers, and laterally self trapped by a nonlinear Kerr effect to yield a plasmon-soliton. The tight field confinement of this scheme is important for potential excitation of plasmon-solitons by a low power continuous wave optical source, which is further assisted by the expected small group velocity of the plasmon-soliton. In this letter we present for the first time both an analysis of TM spatial solitons, using the Non- linear Schrödinger Equation (NLSE), where the full vectorial field is considered, as well as the prediction and characteristics of SPP based solitons.
    [Show full text]
  • Soliton and Rogue Wave Statistics in Supercontinuum Generation In
    Soliton and rogue wave statistics in supercontinuum generation in photonic crystal fibre with two zero dispersion wavelengths Bertrand Kibler, Christophe Finot, John M. Dudley To cite this version: Bertrand Kibler, Christophe Finot, John M. Dudley. Soliton and rogue wave statistics in supercon- tinuum generation in photonic crystal fibre with two zero dispersion wavelengths. The European Physical Journal. Special Topics, EDP Sciences, 2009, 173 (1), pp.289-295. 10.1140/epjst/e2009- 01081-y. hal-00408633 HAL Id: hal-00408633 https://hal.archives-ouvertes.fr/hal-00408633 Submitted on 17 Apr 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. EPJ manuscript No. (will be inserted by the editor) Soliton and rogue wave statistics in supercontinuum generation in photonic crystal ¯bre with two zero dispersion wavelengths Bertrand Kibler,1 Christophe Finot1 and John M. Dudley2 1 Institut Carnot de Bourgogne, UMR 5029 CNRS-Universit¶ede Bourgogne, 9 Av. A. Savary, Dijon, France 2 Institut FEMTO-ST, UMR 6174 CNRS-Universit¶ede Franche-Comt¶e,Route de Gray, Besan»con, France. Abstract. Stochastic numerical simulations are used to study the statistical prop- erties of supercontinuum spectra generated in photonic crystal ¯bre with two zero dispersion wavelengths.
    [Show full text]
  • Soliton Molecules, Rational Positon Solution and Rogue Waves for the Extended Complex Modi Ed Kdv Equation
    Soliton Molecules, Rational Positon Solution and Rogue Waves for the Extended Complex Modied KdV Equation Lin Huang Hangzhou Dianzi University Nannan Lv ( [email protected] ) Hangzhou Dianzi University School of Science Research Article Keywords: Darboux transformation, Soliton molecule, Positon solution, Rational positon solution, Rogue waves solution Posted Date: May 11th, 2021 DOI: https://doi.org/10.21203/rs.3.rs-434604/v1 License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License Version of Record: A version of this preprint was published at Nonlinear Dynamics on August 5th, 2021. See the published version at https://doi.org/10.1007/s11071-021-06764-x. Soliton molecules, rational positons and rogue waves for the extended complex modified KdV equation Lin Huang ∗and Nannan Lv† School of Science, Hangzhou Dianzi University, 310018 Zhejiang, China. May 6, 2021 Abstract We consider the integrable extended complex modified Korteweg–de Vries equation, which is generalized modified KdV equation. The first part of the article considers the construction of solutions via the Darboux transformation. We obtain some exact solutions, such as soliton molecules, positon solutions, rational positon solutions, and rogue waves solutions. The second part of the article analyzes the dynamics of rogue waves. By means of the numerical analysis, under the standard decomposition, we divide the rogue waves into three patterns: fundamental pattern, triangular pattern and ring pattern. For the fundamental pattern, we define the length and width of the rogue waves and discuss the effect of different parameters on rogue waves. Keywords: Darboux transformation, Soliton molecule, Positon solution, Rational positon so- lution, Rogue waves solution.
    [Show full text]
  • Andaman Sea by Global Ocean Associates Prepared for the Office of Naval Research - Code 322PO
    An Atlas of Oceanic Internal Solitary Waves (May 2002) The Andaman Sea by Global Ocean Associates Prepared for the Office of Naval Research - Code 322PO Andaman Sea Overview The Andaman Sea is located along the eastern side of the Indian Ocean between the Malay Peninsula and the Andaman and Nicobar Islands (Figure 1). It is a deep-water sea with exits to the Indian Ocean (to the west) and the Strait of Malacca (to the south). Figure 1. Bathymetry of Andaman Sea [Smith and Sandwell, 1997]. 485 An Atlas of Oceanic Internal Solitary Waves (May 2002) The Andaman Sea by Global Ocean Associates Prepared for the Office of Naval Research - Code 322PO Observations As far back as the mid-19th century surface manifestations of solitons have been observed consisting of strong bands of sea surface roughness. These bands were referred to as "ripplings", due to their mistaken association with rip tides. A description of such bands can be found in the book of Maury [1861] published in 1861 and which is quoted in Osborne and Burch [1980]: "In the entrance of the Malacca Straits, near Nicobar and Acheen Islands, and between them and Junkseylon, there are often strong ripplings, particularly in the southwest monsoon; these are alarming to persons unacquainted, for the broken water makes a great noise when the ship is passing through the ripplings at night. In most places ripplings are thought to be produced by strong currents, but here they are frequently seen when there is no perceptible current…so as to produce an error in the course and distance sailed, yet the surface if the water is impelled forward by some undiscovered cause.
    [Show full text]
  • Soliton-Guided Quantum Information Processing
    Chapter 1 Soliton-Guided Quantum Information Processing Ken Steiglitz Computer Science Department Princeton University Princeton NJ 08544 Abstract We describe applications of solitons and soliton collisions to the transport, transfer, and beam-splitting of qubits carried by optical photons. The transport and transfer realize the “flying qubits” necessary for quantum information processing, and the beam-splitting leads, in theory, to an implementation of quantum computing using linear optics. These proposed applications are embedded in a uniform optical fiber and require no special device fabrication, no cooling, and no vacuum. The pioneering papers of Feynman [1] and Deutsch [2] in the 1980s sparked the rapid development of the field of quantum information processing. The theoretical and experimentalprogress has been remarkable, with the development, for example, of quantum error correction and a fast algorithm for factoring, and the exploration of a wide variety of physical implementations. In the latter category, the optical photon as the carrier of a qubit has played an important role in the experimental demonstra- tion of quantum cryptography and other important applications to communications and information processing. At the same time there has been tremendous progress in our understanding of classical nonlinear waves, and, in particular, solitons in op- tical fibers. In this chapter we will explore what role solitons and soliton collisions might play in the development of quantum information processing with optical pho- tons. For a more detailed account, the reader is referred to [3, 4, 5], from which the material in this chapter was drawn. 1 2 Steiglitz 1.1 Photon trapping A pulse traveling down a fiber forms a soliton when the dispersion, which tends to widen the pulse, is counterbalanced by the nonlinear Kerr effect, whereby the elec- tric field changes the index of refraction of the material.
    [Show full text]
  • Mysterious Underwater Waves
    FEATURE | PROF. LEO MAAS, NIOZ ROYAL NETHERLANDS INSTITUTE FOR SEA RESEARCH, THE NETHERLANDS Uncharted Reality? Mysterious Underwater Waves In addition to surface waves generated by wind, tidal forces or, in case of tsunamis, seismic events, there exists a less well-known class of bigger waves under water. These underwater waves (internal waves) are generated by the same forces, but owe their existence to the ocean’s non-uniform density distribution. This paper introduces various underwater waves that appear on interfaces or in continuous density stratification. In uniformly-stratified seas, these waves differ from surface waves in nearly every conceivable aspect, creating underwater ‘storms’ at certain hotspots, which may possibly impact natural and man-made structures, and which may transport heat and material vertically. The ocean’s salt and heat content, which its boundary. While ocean and atmosphere approximately a factor thousand, the ratio of determines its density, vary with depth. are dynamically quite similar, we never see the cross-thermocline density difference to This stable stratification supports underwater such waves in the ocean though, because mean density. Therefore, interfacial waves waves. Due to slight differences in density, the ocean is opaque. Light penetrates a mere attain larger amplitudes (up to hundreds of these waves easily displace fluid parcels hundred metres, leaving the remainder of metres), have longer periods (longer than vertically over hundreds of metres. the ocean literally and figuratively in the dark. 10 minutes, say) and have wavelengths (up Depending on the local rate with which But oceanographic instruments ‘visualise to kilometres) that are short compared to density increases with depth, they propagate the invisible’.
    [Show full text]
  • Soliton Molecules and Optical Rogue Waves
    SolitonSoliton MoleculesMolecules andand OpticalOptical RogueRogue WavesWaves Benasque, October 2014 Fedor Mitschke Universität Rostock, Institut für Physik [email protected] Part V a Optical Supercontinuum WhyWhy generategenerate broadbandbroadband lightlight fromfrom lasers?lasers? Lasers are the ultimate narrowband (long temporal coherence) sources, but broadband (temporally incoherent) light also has many uses, e.g.: coherence tomography metrology Spectral power density of thermal light is given by Planck‘s law: atat any any frequency frequency ,, therethere is is a a single single parameter parameter T T WhyWhy generategenerate broadbandbroadband lightlight fromfrom lasers?lasers? source: www.nktphotonics.com ... to overcome the Planck barrier! HowHow toto generategenerate broadbandbroadband lightlight fromfrom laserslasers The standard setup: high laser power highly nonlinear fiber microstructuredmicrostructured fiberfiber light-guiding core holey fiber and the tarsal claw of an ant No universal definition of supercontinuum. » Spectral width close to one octave or beyond « J. K. Ranka et al., Opt. Lett. 25, 25 (2000) Supercontinuum generated in 75 cm of microstructured fiber, covering ca. two octaves. Inset: initial 100-fs pulse HowHow doesdoes thethe structurestructure arise?arise? • For fs pump pulses: soliton fission, subsequent soliton frequency shift For ps and longer pump: length scale for fission to arise may be longer than fiber. Dominant process is modulation instability sideband generation. higher-order solitons Length scale: N = 3 HowHow doesdoes thethe structurestructure arise?arise? • For fs pump pulses: soliton fission, subsequent soliton frequency shift For ps and longer pump: length scale for fission to arise may be longer than fiber. Dominant process is modulation instability sideband generation. • Then, all kinds of interactions: Raman shifting, four-wave mixing, coupling between solitons and radiation, etc.
    [Show full text]